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Positivity

, Volume 15, Issue 2, pp 185–197 | Cite as

Positive almost Dunford–Pettis operators and their duality

  • Belmesnaoui Aqzzouz
  • Aziz Elbour
  • Anthony W. Wickstead
Article

Abstract

We study some properties of almost Dunford–Pettis operators and we characterize pairs of Banach lattices for which the adjoint of an almost Dunford–Pettis operator inherits the same property and look at conditions under which an operator is almost Dunford–Pettis whenever its adjoint is.

Keywords

Almost Dunford–Pettis operator Order continuous norm Positive Schur property KB-space 

Mathematics Subject Classification (2000)

46A40 46B40 46B42 

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References

  1. 1.
    Aliprantis, C.D., Burkinshaw, O.: Positive operators. Springer, Dordrecht (2006). Reprint of the 1985 original. MR 2262133Google Scholar
  2. 2.
    Aqzzouz B., Bourass K., Elbour A.: Some generalizations on positive Dunford–Pettis operators. Results Math. 54(3–4), 207–218 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Dodds, P.G., Fremlin, D.H.: Compact operators in Banach lattices. Israel J. Math. 34(4), 287–320 (1979). (1980) MR 570888 (81g:47037)Google Scholar
  4. 4.
    Josefson B.: Weak sequential convergence in the dual of a Banach space does not imply norm convergence. Ark. Mat. 13, 79–89 (1975) MR 0374871 (51 #11067)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Meyer-Nieberg, P.: Banach Lattices. Universitext, Springer-Verlag, Berlin (1991). MR 1128093 (93f:46025)Google Scholar
  6. 6.
    Nakano, H.: Über die Charakterisierung des allgemeinen C-Raumes. Proc. Imp. Acad. Tokyo 17, 301–307 (1941) (German). MR 0014175 (7,249h)Google Scholar
  7. 7.
    Nakano, H.: Über normierte teilweisegeordnete Moduln. Proc. Imp. Acad. Tokyo 17, 311–317 (German) (1941). MR 0014174 (7,249g)Google Scholar
  8. 8.
    Nissenzweig A.: W* sequential convergence. Israel J. Math. 22(3–4), 266–272 (1975) MR 0394134 (52 #14939)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Wickstead A.W.: An isomorphic version of Nakano’s characterization of C 0(Σ). Positivity 11(4), 609–615 (2007) MR 2346446 (2008g:46032)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Wnuk, W.: Some characterizations of Banach lattices with the Schur property. Rev. Mat. Univ. Complut. Madrid 2(suppl.), 217–224 (1989). Congress on Functional Analysis (Madrid, 1988). MR 1057221 (91f:46033)Google Scholar
  11. 11.
    Wnuk W.: Banach lattices with properties of the Schur type—a survey. Confer. Sem. Mat. Univ. Bari 249, 25 (1993) MR 1230964 (94h:46031)MathSciNetGoogle Scholar
  12. 12.
    Wnuk W.: Remarks on J R Holub’s paper concerning Dunford–Pettis operators. Math. Japon. 38(6), 1077–1080 (1993) MR 1250331 (94i:46028)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Wnuk W.: Banach lattices with the weak Dunford–Pettis property. Atti Sem. Mat. Fis. Univ. Modena 42(1), 227–236 (1994) MR 1282338 (95g:46034)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Zaanen, A.C.: Riesz spaces. II, North-Holland Mathematical Library, vol. 30. North-Holland Publishing Co., Amsterdam (1983). MR 704021 (86b:46001)Google Scholar

Copyright information

© Birkhäuser / Springer Basel AG 2010

Authors and Affiliations

  • Belmesnaoui Aqzzouz
    • 1
  • Aziz Elbour
    • 2
  • Anthony W. Wickstead
    • 3
  1. 1.Département d’Economie, Faculté des Sciences Economiques, Juridiques et SocialesUniversité Mohammed V-SouissiSalaAljadidaMorocco
  2. 2.Département de Mathématiques, Faculté des SciencesUniversité Ibn TofailKénitraMorocco
  3. 3.Pure Mathematics Research CentreQueens University BelfastBelfastNorthern Ireland, UK

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